Circuit Theory/Phasors/Examples/example11

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Complex 2R, L, C, voltage circuit

Given that the voltage source is defined by  V_s(t) = cos(t) and the current source is defined by  I_s(t) = 2sin(t-\frac{\pi}{3}), find all other voltages, currents and check power.

Circuit marked up for analysis

Knowns, Unknowns and Equations[edit]

Knowns:  V_s, R_1, R_2, L, C, I_s
Unknowns:  v_1, v_2, v_C, v_L, i_1, i_2, i_C, i_L
Equations:
 v_1 = R_1 * i_1
 v_2 = R_2 * i_2
 i_C = C * {d \over dt}v_C
 v_L = L * {d \over dt}i_L
 i_1 + I_s - i_2 -i_C = 0
 i_c - I_s - i_L = 0
 v_1 + v_2 - V_s = 0
 v_C + v_L - v_2 = 0

Solutions and Analysis[edit]

numeric solution calculated in the phasor domain
attempt at a Laplace solution
simulation at circuitlab